Answer: The bases do not have the same area because the volume of the cylinder is not three times the volume of the cone, given the same heights.
Step-by-step explanation: Ideally, the volume of a cone should be a third of the volume of a cylinder and vice versa, if they have the same height and the same base radius. This means, from this question, the volume of both figures have been provided which are as follows;
Volume of cylinder = πr²h
2512 = πr²h
Also, the volume of a cone is given as;
Volume of cone = 1/3(πr²h)
1256 = 1/3(πr²h)
However, we can observe here that 1256 is not a third of 2512.
Also, the radii of both shapes must be equal for them both to have the same area. The radius of the cylinder is derived as follows;
Volume = πr²h
2512 = 3.14 x r² x 12
2512/(3.14 x 12) = r²
2512/37.68 = r²
66.67 = r²
Add the square root to both sides of the equation
√66.67 = √r²
8.17 = r (radius of cylinder equals 8.17 inches)
Likewise the radius of the cone is derived as follows;
Volume = 1/3(πr²h)
1256 x 3 = 3.14 x r² x 12
3768 = 37.68 x r²
3768/37.68 = r²
100 = r²
Add the square root sign to both sides of the equation
√100 = √r²
10 = r (radius of cone equals 10 inches)
Having different radii, the base area of both shapes cannot be the same. Hence the volume of the cylinder is not three times that of the cone, given the same heights.
